91Ó°ÊÓ

Prove in a similar manner that $$ D_{x} \cos x=-\sin x $$

A point describes a circle of radius \(200 \mathrm{ft}\). at the rate of \(20 \mathrm{ft}\). a seeond. How fast is its projection on a fixed diameter travelling when the distance of the point from the diameter is 100 ft. ?

In the accompanying figure determine the following limits when \(\alpha\) approaches 0 : $$ \lim \frac{A R}{M P} $$

A wall 27 ft. high is 64 ft. from a house. Find the length of the shortest ladder that will reach the house if one end rests on the ground outside the wall.

Differentiate the following functions. $$ u=\cos a x $$ $$ \frac{d u}{d x}=-a \sin a x $$

Differentiate the following functions. $$ y=\cos ^{2} x . \quad \frac{d y}{d x}=-2 \sin x \cos x $$

A flywheel 15 ft. in diameter is making 3 revolutions a second. The sun casts horizontal rays which lie in or are parallel to the plane of the flywheel. A small protuberance on the rim of the wheel throws a shadow on a vertical wall. How fast is the shadow moving when it is \(4 \mathrm{ft}\). above the level of the axle?

Plot the spiral, $$r=\frac{1}{\theta}$$ Show thät it has an asymptote parallel to the prime vector. Suggestion. Consider the distance of a point \(P\) of the curve from the prime direction, and find the limit of this distance when \(\theta\) approaches 0 . Determine the angle at which the radius vector corresponding to \(\theta=\pi / 2\) meets this curve.

The equal sides of an isosceles triangles are each 8 in. long, the base being variable. Show that the triangle of maximum area is the one which has a right angle. Take one of the base angles as the independent variable, \(\phi\).

A flywheel 15 ft. in diameter is making 3 revolutions a second. The sun casts horizontal rays which lie in or are parallel to the plane of the flywheel. A small protuberance on the rim of the wheel throws a shadow on a vertical wall. How fast is the shadow moving when it is \(4 \mathrm{ft}\). above the level of the axle?